Active contours and minimal paths have been extensively studied theoretical tools for image segmentation. The recent geodesically linked active contour model, which basically consists in a set of vertices connected by paths of minimal cost, blend the benefits of both concepts. This makes up a closed piecewise-defined curve, over which an edge or region energy functional can be formulated. As an important shortcoming, the geodesically linked active contour model in its initial formulation does not guarantee to represent a simple curve, consistent with respect to the purpose of segmentation. In this paper, we propose to extract a similarly piecewise-defined curve from a set of possible paths, such that the resulting structure is guaranteed to represent a relevant closed curve. Toward this goal, we introduce a global constraint penalizing excessive overlap between paths.